In ARM we discuss how you can go back and forth between logit and probit models by dividing by 1.6. Or, to put it another way, logistic regression corresponds to a latent-variable model with errors that are approximately normally distributed with mean 0 and standard deviation 1.6. (This is well known, it’s nothing original with our book.) Anyway, John Cook discusses the approximation here.

Nice link. This has also cropped up in meta-analysis, see Chinn Stat Med 2000, which matches moments.